# An Unsolvable Integral (according to Matlab)

• MATLAB
• ILens
In summary, the conversation discusses the inability to solve a certain integral in Matlab and the successful calculation of the same integral in Mathcad and Mathematica. The use of the theorem of residues is suggested as a possible method for solving the integral. The conversation also references a related integral found in G & R 5th edition, CD version, and provides the solution for another related integral calculated using Mathematica.
ILens
Hello!

I have the following problem: I cannot solve the integral below by the means of Matlab.
$$\int_{-\infty}^{\infty} \frac{e^{-t^2}}{\left(2-t\right)^2 + 16} dt$$​
When I write the following in Matlab
Code:
>> syms t;
>> y = exp(-t^2) / (16 + (2 - t)^2);
>> int(y, t, -inf, inf)
it gives me the output
Code:
Warning: Explicit integral could not be found.
> In sym.int at 58

ans =

int(exp(-t^2)/(16+(2-t)^2),t = -Inf .. Inf)

I managed to calculate the integral by the means of both Mathcad and Mathematica. Mathcad gave 0.088 as an answer (I had to explicitly specify "Infinite Limit" as a method). Mathematica gave me 0.0880741, I used the NIntegrate function:

$$\mbox{NIntegrate}\left[\frac{e^{-t^2}}{\left(2-t\right)^2 + 16}, \left\{t, -\infty, \infty \right\} \right]$$​

Does anyone have an idea, how I can solve this integral in Matlab? What do Mathcad and Mathematica use in order to solve it?

Thanks!

I think it/they use(s) the theorem of residues.The integrand has simple poles at $2\mp 4i$.

Daniel.

I didn't find this integral in G & R 5-th edition,CD version

$$\int_{0}^{\infty} \frac{e^{-ax^{2}+bx+c}}{x^{2}+d^{2}} \ dx$$

,but this one was

$$\int_{0}^{\infty} \frac{e^{-\mu^{2}x^{2}}}{x^{2}+b^{2}} \ dx$$

Daniel.

Last edited:
dextercioby said:
G & R 5-th edition,CD version

Could you please explain what "G & R" is?

Daniel.

Here it is,courtesy of Mathematica,a closely related integral.

$$\int_{-\infty}^{+\infty} \frac{e^{-x^{2}}}{(2-x)^{2}+4} \ dx =\frac{\sqrt{\pi}}{6}\left[3\sqrt{\pi}\cos 8-12 \ _{1}F_{2}\left(1,\frac{3}{4},\frac{5}{4};-16\right) +64 \ _{1}F_{2}\left(1,\frac{5}{4},\frac{7}{4};-16\right)\right]$$

Daniel.

## 1. What is an unsolvable integral?

An unsolvable integral is a type of mathematical problem that cannot be solved using traditional methods. It involves finding the area under a curve that cannot be expressed in terms of known functions.

## 2. Why is this integral considered unsolvable?

This integral is considered unsolvable because there is no known algebraic expression or series expansion that can accurately represent the function. Therefore, it cannot be integrated using traditional methods such as substitution, integration by parts, or partial fractions.

## 3. Can Matlab solve an unsolvable integral?

No, Matlab cannot solve an unsolvable integral. It is a powerful tool for solving various mathematical problems, but even it has its limitations. If there is no known analytical solution for the integral, Matlab will not be able to find one.

## 4. Are there other methods for approximating the value of an unsolvable integral?

Yes, there are other methods for approximating the value of an unsolvable integral. These include numerical methods such as the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods use a series of approximations to calculate the integral to a desired level of accuracy.

## 5. Can an unsolvable integral be solved using advanced mathematical techniques?

It is possible for an unsolvable integral to be solved using advanced mathematical techniques such as contour integration, differential equations, or complex analysis. However, these methods require a deep understanding of advanced mathematics and are not always feasible for every integral.

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